Implementation, application and analysis of selected algorithms for grid-based discretization and fractal analysis of virtual 3D structures

PhD topic D7

Implementation, application and analysis of selected algorithms for grid-based discretization and fractal analysis of virtual 3D structures

Description:

In the mathematical literature, various definitions of the "fractal dimension" of a point set can be found. For some of these definitions, there exist algorithms to estimate the corresponding fractal dimension of a given object in space. A typical example is the "box-counting dimension", which can be estimated by counting the numbers of cells hit by the given object when it is covered with grids of varying resolution. Fractal dimension and related parameters are interesting values for the characterization of natural objects with a complex structure (e.g., porous media, plant root or shoot systems, coastlines, galaxies).
In a practical part of this Ph.D. project, several algorithms for grid-related discretization and fractal dimension estimation (including statistical confidence information) shall be implemented in the programming language Java or XL, with the aim to build a plugin for the software tool GroIMP (Kniemeyer 2008). These algorithms shall be applied to selected artificial and natural structures represented in GroIMP, including virtual botanical tree crowns and root systems. The algorithms shall be compared in terms of applicability, accuracy, time and space efficiency, senseful interpretability of results, and mathematical/statistical soundness. Furthermore, a comparison with existing implementations of fractal dimension estimators from the literature and from the web shall be carried out.

Literature:

Edgar, G. A. (1992): Measure, Topology, and Fractal Geometry. Springer, New York etc. (230 pp.).
Falconer, K. J. (1990): Fractal Geometry. Mathematical Foundations and Applications. Wiley, Chichester (dt.: Fraktale Geometrie. Spektrum Akad. Verl., Heidelberg etc. 1993).
GroIMP: http://www.grogra.de
Kniemeyer, O. (2008): Design and Implementation of a Graph Grammar Based Language for Functional-Structural Plant Modelling. Ph.D. thesis, University of Technology at Cottbus. [http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:kobv:co1-opus-5937]
Oppelt, A. L.; Kurth, W.; Dzierzon, H.; Jentschke, G.; Godbold, D. L. (2000): Structure and fractal dimensions of root systems of four co-occurring fruit tree species from Botswana. Ann. For. Sci., 57 (2000), 463-475. http://www.uni-forst.gwdg.de/~wkurth/cb/html/frac_a.pdf.gz